General structural model Part 1: Covariance structure and identification. Psychology 588: Covariance structure and factor models

Transcription

1 General structural model Part 1: Covariance structure and identification Psychology 588: Covariance structure and factor models

2 Latent variables 2 Interchangeably used: constructs --- substantively defined concepts common factors or components --- something uncorrelated with measurement errors (cf. case of cause indicators) true scores in measurement theory Generality vs. specificity --- latent variables are operationally defined to imply a set of indicators (but the reverse is not true), i.e., more generalizable than what s intended by each indicator Good indicators represent distinctive aspects of a defined concept reasonably inter-correlated (i.e., internally reliable) since they possess some common characteristics factorially simple; ideally uni-factorial

3 General structural models 3 General structural equation models combine: measurement models that operationally define a set of theoretical concepts, allowing for fallible measurement, with a path model that explains complex relationships among a set of (mostly) latent variables All forms of models considered so far are special cases of GM (e.g., path models only with observed variables, CFA, MIMIC, etc.) --- rules and principles considered so far apply only to a part of GM, and so we will consider ways of combining them as well

4 The general SE model subsumes most multivariate (causal) models, including MANOVA, discriminant function analysis, multivariate regression, canonical correlation analysis (CCA) The equivalence is about the model form, not about what optimization function is used The above listed models can be considered as special cases of CCA, which uses the OLS like loss function (i.e., maximum accounted variance of manifest variables) This type of variance maximization tends to fit more of variances than covariances --- cf. principal component analysis vs. common factor analysis Component model type of approach to SEM --- Partial Least Squares (Wold, 1974, European Economic Review, 67-86) and Generalized Structured Component Analysis (Hwang & Takane, 2004, Psychometrika, 81-99)

5 The model form 5 By combining measurement models and a path model, we mean: y x y 1 η BηΓξζ IB Γξ ζ y Ληε Λ IB Γξ ζ ε x Λξδ 1 where we have 4 parameter matrices of regression weights (B, Γ, Λ y, Λ x ) and 4 covariance matrices for exogenous latent variables (Φ, Ψ, Θ ε, Θ δ ) --- any cause indicators included in ξ Based on this structural representation, we build covariance structure of observed variables (y and x)

6 Implied covariance matrix 6 Σθ Σyy θ Σyx θ Σxy θ Σxx θ Λ y I B ΓΦΓ Ψ I B Λ y Θ Λy I B ΓΦΛ x 1 ΛΦΓ IB Λ Λ ΦΛ Θ x y x x Compare to the case only with observed variables: I B ΓΦΓ Ψ I B I B ΓΦ ΦΓ I B Φ 1 What if some ε are correlated with δ?

7 Identification 7 θ is globally identified if no vector θ 1 and θ 2 exist such that: Σθ Σθ, θθ If any element of θ is unidentifiable, estimates of all others, if any, shouldn t be interpreted Identifiability, as before, means parameters can be identified by algebraic form but doesn t mean will be numerically E.g., γ 11 is algebraically solved to be cov x, y cov x, x and sample estimate of cov(x 2, x 1 ) is very close to What we ve learned so far about identification equally applies here but only for parts of GM, not as a whole

8 t-rule Number of free parameters in θ does not exceed distinctive elements in S: 1 t pq pq1 2 Necessary, not sufficient S How many distinctive elements in

9 Two-step rule 9 Step ignore all paths between latent variables, treat them as simply mutually correlated factors, and use any feasible identification rule for CFA (with fully unconstrained Φ) Step if passes step 1, ignore all measurement relationships and use any feasible identification rule for SEM with observed variables Any cause indicators can be treated as latent variables with loading of 1 and no measurement error --- equivalent to simply including them in ξ Sufficient, not necessary; so, there will be identifiable models that do not pass two-step rule --- local identification will be useful in such cases, though it s empirical and fallible See Fig. 8.3 (p. 329), 8.4 (p. 330) and 8.5 (p. 333)

10 MIMIC rule 10 Multiple Indicators and MultIple Causes (MIMIC) model --- a latent variable is measured with multiple (reflective) indicators and caused by multiple observed variables (formative or cause indicators, or covariates) A sufficient condition per η for identification of a MIMIC model: p 2, q 1 X g Eta 1 a Y1 e1 Y2 e2 S z 22

11 For cases when m > 1, if all cause indicators affect all η s and each η is measured with at least 2 indicators, the 2-step rule applies; how? In general, too limited model condition in that no path modeling allowed between latent variables